A362610 Number of integer partitions of n having a unique part of least multiplicity.

0, 1, 2, 2, 4, 5, 7, 10, 13, 16, 23, 30, 35, 50, 61, 73, 95, 123, 139, 187, 216, 269, 328, 411, 461, 594, 688, 836, 980, 1211, 1357, 1703, 1936, 2330, 2697, 3253, 3649, 4468, 5057, 6005, 6841, 8182, 9149, 10976, 12341, 14508, 16447, 19380, 21611, 25553, 28628

Offset

0,3

Comments

Alternatively, these are partitions with a part appearing fewer times than each of the others.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Formula

G.f.: Sum_{m>=2} (Sum_{j>=1} x^(j*(m-1))/(1 + x^(j*m)/(1 - x^j))) * (Product_{j>=1} (1 + x^(j*m)/(1 - x^j))). - Andrew Howroyd, May 04 2023

Example

The partition (3,3,2,2,2,1,1,1) has least multiplicity 2, and only one part of multiplicity 2 (namely 3), so is counted under a(15).
The a(1) = 1 through a(8) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (221)    (33)      (322)      (44)
                    (211)   (311)    (222)     (331)      (332)
                    (1111)  (2111)   (411)     (511)      (422)
                            (11111)  (3111)    (2221)     (611)
                                     (21111)   (4111)     (2222)
                                     (111111)  (22111)    (5111)
                                               (31111)    (22211)
                                               (211111)   (41111)
                                               (1111111)  (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)

Mathematica

Table[Length[Select[IntegerPartitions[n], Count[Length/@Split[#], Min@@Length/@Split[#]]==1&]], {n, 0, 30}]

Prog

(PARI) seq(n) = my(A=O(x*x^n)); Vec(sum(m=2, n+1, sum(j=1, n, x^(j*(m-1))/(1 + if(j*m<=n, x^(j*m)/(1-x^j) )) + A)*prod(j=1, n\m, 1 + x^(j*m)/(1 - x^j) + A)), -(n+1)) \\ Andrew Howroyd, May 04 2023

Crossrefs

For parts instead of multiplicities we have A002865, ranks A247180.

For median instead of co-mode we have A238478, complement A238479.

These partitions have ranks A359178.

For mode complement of co-mode we have A362607, ranks A362605.

For mode instead of co-mode we have A362608, ranks A356862.

The complement is counted by A362609, ranks A362606.

A000041 counts integer partitions.

A275870 counts collapsible partitions.

A359893 counts partitions by median.

A362611 counts modes in prime factorization, co-modes A362613.

A362614 counts partitions by number of modes, co-modes A362615.

Cf. A008284, A053263, A098859, A237984, A240219, A304442, A327472, A353863, A353864, A353865, A362612.

Sequence in context: A098859 A364345 A239455 * A363260 A195012 A333192

Adjacent sequences: A362607 A362608 A362609 * A362611 A362612 A362613

Keyword

nonn

Author

Gus Wiseman, Apr 30 2023

Status

approved